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# A binary communication system transmits a signal X in the following way: 1 is transmitted if a 0 bit is to be communicated, +1 is transmitted if a 1...

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A binary communication system transmits a signal X in the following way: —1 istransmitted if a 0 bit is to be communicated, +1 is transmitted if a 1 bit is to be communicated. The received signal is Y = X + N. N is noise with zero—mean Normaldistribution with variance 02. Assume that the 0 bits are 4 times as likely as 1 bits. (a) Find the conditional PDF of Y given the input value:1.1%le = +1)&11d fﬁylx = —1Jl (b) The receiver decides a 0 bit was sent if the received value of y hasfYCUlX = —1)P[X = —1l> 1’14le : +1)P[X = +1], and decides the bit was 1 otherwise. Using the results from the previous part,show that this rule is equivalent to: If 1” <2 T1 decide 0, if Y 3 T, decide 1, whereT is some threshold. (c) If 02 = 16, what is the probability that the receiver makes an error given a +1was transmitted? What about if a —1 was transmitted? [d] What is the overall error probability when 02 = 16?